Using the concepts from Section 8.3, we are now ready to define feedback motion planning over configuration spaces or other continuous state spaces. Recall Formulation 4.1, which defined the basic motion planning problem in terms of configuration space. The differences in the current setting are that there is no initial condition, and the requirement of a solution path is replaced by a solution vector field. The formulation here can be considered as a continuous-time adaptation to Formulation 8.1.
A feedback plan, , for Formulation 8.2 is defined as a function , which produces an action for each . A feedback plan can equivalently be considered as a vector field on because each specifies a velocity vector ( specifies zero velocity). Since the initial state is not fixed, it becomes slightly more complicated to define what it means for a plan to be a solution to the problem. Let denote the set of all states from which is reachable. More precisely, a state belongs to if and only if a continuous path exists for which and for some . This means that a solution path exists from for the ``open-loop'' motion planning problem, which was considered in Chapter 4.